An introduction to the general theory of second order parabolic differential equations, which model many time-dependent physical systems. The book studies the existence, uniqueness and regularity of solutions to a variety of problems with Dirichlet boundary conditions and general linear and nonlinear boundary conditions by means of "a priori" estimates. The first seven chapters give a description of the linear theory and are suitable for a graduate course on partial differential equations. The last eight chapters cover the nonlinear theory for smooth solutions. They include much of the author's research and are aimed at researchers in the field. A feature is the emphasis on time-varying domains.